A polyhedron is regular if its faces and vertex angles are all regular.

There are nine regular polyhedra: five Platonic solids, and four Kepler-Poinsot polyhedra.

In Maple, one can define a regular polyhedron by using the command RegularPolyhedron(gon, [m, n], o, r) where gon is the name of the polyhedron to be defined, $\left[m\,n\right]$ the Schlafli symbol, o the center of the polyhedron.

When r is a positive number, it specifies the radius of the circum-sphere. When r is an equation, the left-hand side is one of radius, side, mid_radius, or in_radius, and the right-hand side specifies the radius of the circum-sphere, the side, the mid-radius or the in-radius (respectively) of the polyhedron to be constructed.

The value of $\left[m\,n\right]$ can be one of the following:

Another way to define a regular polyhedron is to use the command PolyhedronName(gon, o, r) where PolyhedronName is one of tetrahedron, octahedron, hexahedron, cube, icosahedron, dodecahedron, GreatStellatedDodecahedron, SmallStellatedDodecahedron, GreatIcosahedron, or GreatDodecahedron.

To access the information relating to a regular polyhedron gon, use the following function calls:

$\mathrm{with}\left(\mathrm{geom3d}\right)\:$

$\mathrm{tetrahedron}\left(t\,\mathrm{point}\left(o\,1\,2\,3\right)\,3\right)$

$t$

$\mathrm{area}\left(t\right)$

$24\sqrt{3}$

$\mathrm{center}\left(t\right)$

$o$

$\mathrm{faces}\left(t\right)$

$\left[\left[\left[1+\sqrt{3}\,2+\sqrt{3}\,\sqrt{3}+3\right]\,\left[1+\sqrt{3}\,2-\sqrt{3}\,-\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2+\sqrt{3}\,-\sqrt{3}+3\right]\right]\,\left[\left[1+\sqrt{3}\,2+\sqrt{3}\,\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2-\sqrt{3}\,\sqrt{3}+3\right]\,\left[1+\sqrt{3}\,2-\sqrt{3}\,-\sqrt{3}+3\right]\right]\,\left[\left[1+\sqrt{3}\,2+\sqrt{3}\,\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2+\sqrt{3}\,-\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2-\sqrt{3}\,\sqrt{3}+3\right]\right]\,\left[\left[1+\sqrt{3}\,2-\sqrt{3}\,-\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2-\sqrt{3}\,\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2+\sqrt{3}\,-\sqrt{3}+3\right]\right]\right]$

$\mathrm{form}\left(t\right)$

$\mathrm{tetrahedron3d}$

$\mathrm{InRadius}\left(t\right)$

$\frac{\sqrt{6}\sqrt{2}\sqrt{3}}{6}$

$\mathrm{MidRadius}\left(t\right)$

$\sqrt{3}$

$\mathrm{radius}\left(t\right)$

$3$

$\mathrm{schlafli}\left(t\right)$

$\left[3\,3\right]$

$\mathrm{sides}\left(t\right)$

$2\sqrt{2}\sqrt{3}$

$\mathrm{vertices}\left(t\right)$

$\left[\left[1+\sqrt{3}\,2+\sqrt{3}\,\sqrt{3}+3\right]\,\left[1+\sqrt{3}\,2-\sqrt{3}\,-\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2+\sqrt{3}\,-\sqrt{3}+3\right]\,\left[-\sqrt{3}+1\,2-\sqrt{3}\,\sqrt{3}+3\right]\right]$

$\mathrm{volume}\left(t\right)$

$8\sqrt{3}$

$\mathrm{RegularPolyhedron}\left(d\,\left[5\,3\right]\,\mathrm{point}\left(o\,0\,0\,0\right)\,1\right)$

$d$

$\mathrm{form}\left(d\right)$

$\mathrm{dodecahedron3d}$

$\mathrm{RegularPolyhedron}\left(\mathrm{d2}\,\left[5\,3\right]\,\mathrm{point}\left(o\,0\,0\,0\right)\,\mathrm{side}=\mathrm{sides}\left(d\right)\right)$

$\mathrm{d2}$

$\mathrm{radnormal}\left(\mathrm{radius}\left(\mathrm{d2}\right)\right)$

$1$